Question

Which term of the Arithmetic Progression $ - 7, - 12, - 17, - 22,..... $ will be $ - 82? $ Is $ - 100 $ any term of the A.P.? Give reason for your answer.

Answer 1

Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is – $ {a_n} = a + (n - 1)d $
Where
 $ {a_n} = $ nth term in the AP
  $ a = $ First term of AP
  $ d = $ Common difference in the series
  $ n = $ Number of terms in the AP
Here we will find the first term, common difference and the given nth term. Also, we will find whether $ ( - 100) $ is any term in the given Arithmetic progression.

Complete step-by-step answer:
Given A.P. is $ - 7, - 12, - 17, - 22,..... $
Here,
 $
  a = - 7 \\
  d = - 12 - ( - 7) = - 12 + 7 = - 5 \\
  {a_n} = - 82 \;
  $
Place the above values in the equation - $ {a_n} = a + (n - 1)d $
 $ \Rightarrow - 82 = - 7 + (n - 1)( - 5) $
Simplify the above equation –
 $ \Rightarrow - 82 = - 7 - 5n + 5 $
Take all the constant terms at one side of the equation and make the unknown term “n” the subject.
 $
   \Rightarrow - 82 + 7 - 5 = - 5n \\
   \Rightarrow - 87 + 7 = -5 n \\
   \Rightarrow - 80 = - 5n \\
    \Rightarrow n= 16
  $
Take “minus” sign common from both the sides of the equations –
 $ \Rightarrow n = 16 $
Therefore, $ 16th $ term of the given arithmetic progression will give $ ( - 82) $
So, the correct answer is “ $ 16th $ term”.

Now, check whether $ ( - 100) $ is any term of the given series-
See $ 16th $ term of the given arithmetic progression will give $ ( - 82) $
Also, the common difference between the terms is $ ( - 5) $
Now, finding the further terms in the series which gives-
 $
   \Rightarrow {t_{17}} = - 82 - 5 = ( - 87) \\
   \Rightarrow {t_{18}} = - 87 - 5 = ( - 92) \\
   \Rightarrow {t_{19}} = - 92 - 5 = ( - 97) \\
   \Rightarrow {t_{20}} = - 97 - 5 = ( - 102) \;
  $
The above equation states that, $ ( - 100) $ is not any of the terms in the given arithmetic progression.
So, the correct answer is “ $ ( - 100) $ is not any of the terms in the given arithmetic progression.”.

Note: you can check $ ( - 100) $ in the given arithmetic progression by using nth term formula.
Know the difference between the geometric progression and the arithmetic progression and apply accordingly. There are two types of sequences and series.
I.Arithmetic progression
II.Geometric Progression.
In arithmetic progression, the difference between the numbers is constant in the series whereas the geometric progression is the sequence in which the succeeding element is obtained by multiplying the preceding number by the constant and the same continues for the series. The ratio between the two remains the same.
The arithmetic series is the sum of all the terms of the arithmetic progression (AP). Where, “a” is the first term and “d” is the common difference among the series.
 $
  {S_n}{\text{ = a + (a + d) + (a + 2d) + (a + 3d) + }}......{\text{ + [a + (n - 1)d]}} \\
  {{\text{S}}_{n{\text{ }}}} = \dfrac{n}{2}[2a + (n - 1)d] \\
  $